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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Covers in free lattices


Authors: Ralph Freese and J. B. Nation
Journal: Trans. Amer. Math. Soc. 288 (1985), 1-42
MSC: Primary 06B25
MathSciNet review: 773044
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Abstract: In this paper we study the covering relation $ (u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $ w \in {\text{FL}}(X)$, finds all the elements which cover or are covered by $ w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in $ {\text{FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0773044-4
PII: S 0002-9947(1985)0773044-4
Article copyright: © Copyright 1985 American Mathematical Society